In this paper, we propose and analyze a $C^1$-virtual element method of high order to solve the Brinkman problem formulated in terms of the stream function. The velocity is obtained as a simple post-process from stream function and a novel strategy is written to recover the fluid pressure. We establish optimal a priori error estimates for the stream function, velocity and pressure with constants independent of the viscosity. Finally, we report some numerical test illustrating the behavior of the virtual scheme and supporting our theoretical results on different families of polygonal meshes.

中文翻译：

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具有压力恢复的流函数公式中 Brinkman 方程的高阶 C1 虚元法

在本文中，我们提出并分析了一种高阶$C^1$-虚元方法来解决用流函数表示的布林克曼问题。速度是通过流函数的简单后处理获得的，并编写了一种新的策略来恢复流体压力。我们建立了流体函数、速度和压力的最佳先验误差估计，其常数与粘度无关。最后，我们报告了一些数值测试来说明虚拟方案的行为，并支持我们在不同多边形网格系列上的理论结果。